Acceleration of the PDHGM on strongly convex subspaces

TL;DR

This paper introduces accelerated primal-dual algorithms that leverage strong convexity in subspaces to improve convergence rates, demonstrated on image processing tasks like denoising and deblurring.

Contribution

It develops variants of Chambolle and Pock's primal-dual method that accelerate convergence on subspaces with strong convexity without requiring full strong convexity.

Findings

Achieves mixed convergence rates of O(1/N^2) and O(1/N)

Effective on image processing problems like total variation denoising

Demonstrates practical improvements in convergence speed

Abstract

We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the efficacy of the proposed methods on image processing problems lacking strong convexity, such as total generalised variation denoising and total variation deblurring.